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Plane Partitions Fermion-Boson Duality Quantum crystals, Kagome lattice, and plane partitions fermion-boson duality Thiago Araujo,1, ∗ Domenico Orlando,1, 2, y and Susanne Reffert1, z 1Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, University of Bern, Sidlerstrasse 5, ch-3012, Bern, Switzerland 2INFN, sezione di Torino and Arnold-Regge Center, via Pietro Giuria 1, 10125 Torino, Italy In this work we study quantum crystal melting in three space dimensions. Using an equivalent description in terms of dimers in a hexagonal lattice, we recast the crystal melting Hamiltonian as an occupancy problem in a Kagome lattice. The Hilbert space is spanned by states labeled by plane partitions, and writing them as a product of interlaced integer partitions, we define a fermion-boson duality for plane partitions. Finally, based upon the latter result we conjecture that the growth operators for the quantum Hamiltonian can be represented in terms of the affine Yangian Y[glb(1)]. I. INTRODUCTION addressed in [9]. Plane partitions can be written in terms of interlaced XXZ diagrams or as lattice fermions, see Random partitions appear in many contexts in mathe- figure (2). The empty partition is a configuration of the matics and physics. This omnipresence is partly explained hexagonal lattice where all positions below the diagonals by the fact that they are part of the core of number theory, are occupied. Additionally, based on the one- and two- the queen of mathematics [1]. It is nonetheless remarkable dimensional cases, a conjecture for the mass gap has been that they appear in a variety of distinct problems { such made, and further numerical analysis has been performed. as the Seiberg{Witten theory, integrable systems, black We present a number of equivalent formulations of the holes, string theory and others [2{6]. 3D quantum crystal melting problem, such as in terms of In two dimensions, random partitions can be successfully particle-hole hopping, a fermionic description on a Kagome realized in terms of fermionic operators living on a chain lattice, and a tensor product representation. We are even- where particles and holes are labeled by half-integers along tually lead to a fermion-boson duality, and to a conjecture the real line [3, 7{9]. In this formalism, the empty partition for an underlying Yangian symmetry. We expect this is equivalent to a configuration where all negative holes (or structure to be instrumental in uncovering a possible hid- better yet, holes on negative positions) are occupied and den integrable structure, suggested by the results in lower all positive holes are vacant. The creation and annihilation dimensions. The plan of this note is as follows. We start with a short review in Section II, where we see how plane partitions are equivalently written in terms of dimers in a hexagonal lattice. FIG. 1: Fermionic fields and Young diagram relation. arXiv:2005.09103v2 [hep-th] 27 Jan 2021 of squares are defined as holes and particles hopping as in figure (1). FIG. 2: 3D Young diagram and its fermions lattice The description above has been used in [9, 10] to study the two-dimensional version of quantum crystal melting. Quite remarkably, with a Jordan-Wigner transformation, As we said before, we want to find an alternative particle- the crystal melting Hamiltonian is shown to be equivalent hole dynamics which is equivalent to the plane partition to the XXZ spin chains with kink boundary conditions, and growth. More explicitly, we search for a statistical sys- it automatically implies the integrability of this problem. tem with states labeled by plane partitions. That is the problem we address in Section III, where we discuss a In this work, we focus on the three-dimensional version construction in terms of particle hopping in a hexagonal of the quantum crystal melting, which can be rephrased in lattice. This formalism is considerably simplified if we terms of random plane partitions, and it has been partly use a dual description which turns out to be the Kagome lattice. We write the new Hamiltonian, and we provide additional details of it in SectionIV. ∗Electronic address: [email protected] Having discussed characterizations of the crystal melting yElectronic address: [email protected] as occupation problems, we try to understand other fun- zElectronic address: sreff[email protected] damental aspects of the system, in particular, its Hilbert 2 space. In SectionV, we write the plane partition states that there is only a finite number of boxes that can be in terms of interlaced integer partitions, and we use this consistently removed, or available places where we can add fact to show that the growth operators have nontrivial a box. coproducts. In addition, we define a novel fermion-boson correspondence for plane partitions states in SectionVI. Finally, based on the action of the growth operators on the bosonized states, we conjecture that this Hamiltonian can represented in terms of the affine Yangian of glb(1). We give some concluding remarks in Section VII. In AppendixA, we give present some necessary details on the 2D fermion- boson duality, in AppendixB we argue the equivalence of the Hamiltonians in the plane-partition language and in the Kagome picture. In AppendixC, we detail basis transformations for various states and in AppendixD, we give amplitudes for transitions between various states. FIG. 4: 1 Box configuration II. PLANE PARTITIONS & HEXAGONAL We want to study the crystal melting Hamiltonian de- DIMERS fined in [9] X We start by describing the well known correspondence H = − J (j ih j + j ih j) between the crystal melting problem and dimers in a hexag- X p 1 (2a) + V qj ih j + p j ih j ; onal lattice, see [9, 11{13] and references therein. The q empty partition (vacuum) is equivalent to an infinity per- fect matching with different staggered boundary conditions that we conveniently write as in the regions I, II and III, see figure (3), defined as p 1 H = − J (Γ + Π) + V q O (Π; Γ) + p O (Π; Γ) : # 2 I = (−π=6; π=2); + q − # 2 II = (π=2; 7π=6); (1) (2b) # 2 III = (−5π=6; −π=6): Using j iy = h j and j iy = h j, one can easily see In the vacuum configuration, the origin is defined as that the Hamiltonian is Hermitian. The two terms multi- the only plaquette where the three phases, or staggered plied by J are kinetic and describe the two flips previously directions, meet. We say that a plaquette with the three mentioned. More specifically, the operator Γ (Π) adds phases is a flippable plaquette. (removes) boxes in all possible ways. Moreover, the poten- tial terms O+(Π; Γ) and O−(Π; Γ) count the number of positive and negative flippable plaquettes, respectively. In other words, they determine the number of places where we can add a box, and the number of boxes that can be removed respecting the plane partition rules. This system has a special point for J = V , the so-called Rokhsar-Kivelson point [14, 15], where it is natural to understand the model as the stochastic quantization of the classical counting of plane partitions [16]. In the continuum limit this corresponds to a quantum critical point [17]. At this point we expect the system to have special symmetry FIG. 3: Empty partition properties and we will focus on this case. Some progress in the understanding of this system has been achieved in [9], where it was shown that the ground state of the system is Adding a box to the vacuum corresponds to the positive given by a sum over all plane partitions, where a partition flip ( ) 7! ( ), as seen in figure (4), and as a matter of of the integer k is weighted by qk=2. If follows that the fact, there is only one flippable plaquette in the vacuum. norm of the ground state is given by Additionally, simple inspection of figure (4) reveals that there are now three plaquettes where we can perform a Y 1 Z ≡ hgroundjgroundi = ; (3) positive flip, and just one plaquette where the negative flip (1 − qn)n ( ) 7! ( ) can be performed. n≥1 It is easy to see that this logic remains the same for general configurations, that is, positive and negative flips which we recognize as the famous MacMahon function: the correspond to adding and removing boxes in a given plane generating function for the number pl(k) of plane partitions partition state. Furthermore, for any lattice configuration, 1 X there is only a finite number of flippable plaquettes, and Z = pl(k)qk : (4) from the plane partition perspective it obviously means k=0 3 In the following sections, we want to represent the Hamil- Furthermore, we set the origin at (j; k) = ~0, and we can 2 tonian (2b) in terms of a statistical mechanics systems of identify a generic point ~x = j~p1 + k~p2,(j; k) 2 Z as in particles and holes, and we start investigating its integra- figure (6). bility properties. More specifically, we define a new form of fermion-boson correspondence and we find connections between this problem and the affine Yangians of glb(1). III. PARTICLE-HOLE BOUND STATES In this section we describe the dynamics of the crystal melting problem in terms of particle hopping in two Bravais FIG. 6: Generic points in the lattice. lattices (2b). In the next section we build a Hamiltonian for this problem using the dual description of this particle-hole Points in the Hole sublattice H can be identified by linear system. combinations of (~p1; ~p2; ~p3) plus the additional offsets First we need to remember that the hexagons above do not define a Bravais lattice, but we can write it as ~ a 1 ~ 1 δ± = p ; ±1 ; δ = a −p ; 0 : (9) a superposition of two triangular Bravais lattices, say H 2 3 3 whose points are labeled by ◦, and P, labeled by •.
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